From Classical to Quantum: Explicit Classical Distributions Achieving Maximal Quantum $f$-Divergence

• URL: http://arxiv.org/abs/2501.14340v1
• Date: Fri, 24 Jan 2025 09:06:13 GMT
• Title: From Classical to Quantum: Explicit Classical Distributions Achieving Maximal Quantum $f$-Divergence
• Authors: Dimitri Lanier, Julien Béguinot, Olivier Rioul,
• Abstract summary: Explicit classical states achieving maximal $f$-divergence are given, allowing for a simple proof of Matsumoto's Theorem. Our methodology is particularly simple as it does not require any elaborate matrix analysis machinery but only basic linear algebra.
• Score: 4.956709222278243
• License:
• Abstract: Explicit classical states achieving maximal $f$-divergence are given, allowing for a simple proof of Matsumoto's Theorem, and the systematic extension of any inequality between classical $f$-divergences to quantum $f$-divergences. Our methodology is particularly simple as it does not require any elaborate matrix analysis machinery but only basic linear algebra. It is also effective, as illustrated by two examples improving existing bounds: (i)~an improved quantum Pinsker inequality is derived between $\chi^2$ and trace norm, and leveraged to improve a bound in decoherence theory; (ii)~a new reverse quantum Pinsker inequality is derived for any quantum $f$-divergence, and compared to previous (Audenaert-Eisert and Hirche-Tomamichel) bounds.

Related papers

• Kochen-Specker for many qubits and the classical limit [55.2480439325792]
  It is shown that quantum and classical predictions converge as the number of qubits is increases to the macroscopic scale. This way to explain the classical limit concurs with, and improves, a result previously reported for GHZ states.
  arXiv  Detail & Related papers  (2024-11-26T22:30:58Z)
• Universal quantum computation using Ising anyons from a non-semisimple Topological Quantum Field Theory [0.058331173224054456]
  We propose a framework for topological quantum computation using newly discovered non-semisimple analogs of topological quantum field theories in 2+1 dimensions. We show that the non-semisimple theory introduces new anyon types that extend the Ising framework.
  arXiv  Detail & Related papers  (2024-10-18T21:03:07Z)
• The Power of Unentangled Quantum Proofs with Non-negative Amplitudes [55.90795112399611]
  We study the power of unentangled quantum proofs with non-negative amplitudes, a class which we denote $textQMA+(2)$. In particular, we design global protocols for small set expansion, unique games, and PCP verification. We show that QMA(2) is equal to $textQMA+(2)$ provided the gap of the latter is a sufficiently large constant.
  arXiv  Detail & Related papers  (2024-02-29T01:35:46Z)
• A Quadratic Speedup in Finding Nash Equilibria of Quantum Zero-Sum Games [102.46640028830441]
  We introduce the Optimistic Matrix Multiplicative Weights Update (OMMWU) algorithm and establish its average-iterate convergence complexity as $mathcalO(d/epsilon)$ to $epsilon$-Nash equilibria. This quadratic speed-up sets a new benchmark for computing $epsilon$-Nash equilibria in quantum zero-sum games.
  arXiv  Detail & Related papers  (2023-11-17T20:38:38Z)
• Quantum $f$-divergences via Nussbaum-Szko{\l}a Distributions and Applications to $f$-divergence Inequalities [0.0]
  This article shows that the quantum $f$-divergence of two states is equal to the classical $f$-divergence of the corresponding Nussbaum-Szkola distributions. The usefulness of the main result is illustrated by obtaining several quantum $f$-divergence inequalities from their classical counterparts.
  arXiv  Detail & Related papers  (2023-08-05T18:03:13Z)
• Optimal lower bounds for Quantum Learning via Information Theory [3.093890460224435]
  We derive optimal lower bounds for quantum sample complexity in both the PAC and models via an information-theoretic approach. We then turn to a quantum analogue of the Coupon Collector problem, a classic problem from probability theory. All the aspects of the Quantum Coupon Collector problem we study rest on properties of the spectrum of the associated Gram matrix.
  arXiv  Detail & Related papers  (2023-01-05T18:55:04Z)
• Quantum Worst-Case to Average-Case Reductions for All Linear Problems [66.65497337069792]
  We study the problem of designing worst-case to average-case reductions for quantum algorithms. We provide an explicit and efficient transformation of quantum algorithms that are only correct on a small fraction of their inputs into ones that are correct on all inputs.
  arXiv  Detail & Related papers  (2022-12-06T22:01:49Z)
• Entanglement and coherence in Bernstein-Vazirani algorithm [58.720142291102135]
  Bernstein-Vazirani algorithm allows one to determine a bit string encoded into an oracle. We analyze in detail the quantum resources in the Bernstein-Vazirani algorithm. We show that in the absence of entanglement, the performance of the algorithm is directly related to the amount of quantum coherence in the initial state.
  arXiv  Detail & Related papers  (2022-05-26T20:32:36Z)
• Monotonic multi-state quantum $f$-divergences [0.0]
  We write down a class of multi-state quantum $f$-divergences and prove that they satisfy the data processing inequality. For two states, this class includes the $(alpha,z)$-R'enyi divergences, the $f$-divergences of Petz, and the measures in citematsumoto2015new as special cases. We conjecture that these multi-state R'enyi divergences have operational interpretations in terms of the optimal error probabilities in asymmetric multi-state quantum state discrimination.
  arXiv  Detail & Related papers  (2021-03-17T20:10:04Z)
• Classical Limits of Unbounded Quantities by Strict Quantization [0.0]
  We introduce the approach first in the simple case of finite systems. We apply this approach to analyze the classical limits of unbounded quantities in bosonic quantum field theories.
  arXiv  Detail & Related papers  (2020-10-14T17:44:51Z)
• From a quantum theory to a classical one [117.44028458220427]
  We present and discuss a formal approach for describing the quantum to classical crossover. The method was originally introduced by L. Yaffe in 1982 for tackling large-$N$ quantum field theories.
  arXiv  Detail & Related papers  (2020-04-01T09:16:38Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.